We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus g ≥ 2 over the rationals whose Jacobian has Mordell-Weil rank g and Picard number greater than one, and which satisfies some additional conditions. This algorithm is then applied to the modular curve Xs(13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.

We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over Q or a quadratic number field. We accomplish this by studying the role of p-adic heights in explicit nonabelian Chabauty.

Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage. Q P ω i } 2g−1 i=0 .

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with Q.

Let X denote a hyperbolic curve over Q and let p denote a prime of good reduction. The third author's approach to integral points, introduced in [Kim2] and [Kim3], endows X(Zp) with a nested sequence of subsets X(Zp)n which contain X(Z). These sets have been computed in a range of special cases [Kim4, BKK, DCW2, DCW3]; there is good reason to believe them to be practically computable in general. In 2012, the third author announced the conjecture that for n sufficiently large, X(Z) = X(Zp)n. This conjecture may be seen as a sort of compromise between the abelian confines of the BSD conjecture and the profinite world of the Grothendieck section conjecture. After stating the conjecture and explaining its relationship to these other conjectures, we explore a range of special cases in which the new conjecture can be verified.